A117256 Triangle T, read by rows, where matrix power T^5 has powers of 5 in the secondary diagonal: [T^5](n+1,n) = 5^(n+1), with all 1's in the main diagonal and zeros elsewhere.

1, 1, 1, -10, 5, 1, 750, -250, 25, 1, -328125, 93750, -6250, 125, 1, 779296875, -205078125, 11718750, -156250, 625, 1, -9741210937500, 2435302734375, -128173828125, 1464843750, -3906250, 3125, 1, 630569458007812500, -152206420898437500, 7610321044921875

Offset

0,4

Comments

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=5, q=5, r=1.

Links

Table of n, a(n) for n=0..30.

Formula

T(n,k) = A117257(n-k)*5^((n-k)*k). T(n,k) = (-1)^(n-k)*5^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(5*j-1) for n>k>=0, with T(n,n) = 1.

Example

Triangle T begins:
1;
1,1;
-10,5,1;
750,-250,25,1;
-328125,93750,-6250,125,1;
779296875,-205078125,11718750,-156250,625,1;
-9741210937500,2435302734375,-128173828125,1464843750,-3906250,3125,1;
Matrix power T^5 has powers of 5 in the 2nd diagonal:
1;
5,1;
0,25,1;
0,0,125,1;
0,0,0,625,1;
0,0,0,0,3125,1; ...

Prog

(PARI) {T(n, k)=local(m=1, p=5, q=5, r=1); prod(j=0, n-k-1, m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

Crossrefs

Cf. A117257 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Sequence in context: A317597 A357423 A320938 * A332837 A050020 A050136

Adjacent sequences: A117253 A117254 A117255 * A117257 A117258 A117259

Keyword

sign,tabl

Author

Paul D. Hanna, Mar 14 2006

Status

approved